HesapHanem

Ellipse Perimeter Calculator

Find the perimeter and area of an ellipse.

How it is calculated

You enter the two semi-axes, a (semi-major) and b (semi-minor), each half of the full width and full height of the ellipse, in the same length unit. The perimeter uses the second Ramanujan approximation P ≈ π × [ 3(a + b) − √( (3a + b)(a + 3b) ) ], and the area is exact: A = π × a × b. Both are rounded to two decimals.

Both axes must be greater than zero. The formula is symmetric, so entering the shorter axis first gives the same answer. When a = b the ellipse is a circle and the formula returns 2πr. The perimeter is in the unit you typed and the area in its square; the result is labelled only as units.

Worked example

An ellipse has a semi-major axis of 5 and a semi-minor axis of 3. First 3(a + b) = 3 × 8 = 24. Then (3a + b)(a + 3b) = 18 × 14 = 252, and its square root is 15.8745. The bracket is 24 − 15.8745 = 8.1255, and multiplying by π gives a perimeter of 25.53 units. The area is π × 5 × 3 = 47.12 units².

Try this example →

Common mistakes

  • Entering the full length and width instead of the semi-axes; the inputs are half of those, so a 10 by 6 ellipse needs 5 and 3.
  • Averaging the axes and using the circle formula 2π × (a + b) / 2; for an elongated ellipse this noticeably misses the perimeter.
  • Treating the perimeter as exact; Ramanujan's formula is a very close approximation, not the elliptic integral itself.
  • Mixing units, such as 5 metres and 300 centimetres, which gives a meaningless shape.
  • Reading the second output as a length: the area is in square units and is not interchangeable with the perimeter.

About this calculator

Unlike a circle's circumference, an ellipse's perimeter has no simple closed-form formula — it requires an elliptic integral. This calculator uses Ramanujan's approximation, P ≈ π[3(a+b) − √((3a+b)(a+3b))], which is accurate to within a tiny fraction of a percent for ordinary ellipses and stays within about 0.4% even for extremely elongated ones, far better than the naive average-radius estimate. It also computes the area, A = π·a·b, which does have an exact formula.

Elliptical shapes appear in orbital mechanics (planetary and satellite orbits are ellipses per Kepler's first law), engineering design of elliptical gears, tanks, and arches, and architecture and landscaping for elliptical gardens, tracks, and domes.

Engineers and architects laying out elliptical structures, students studying Kepler's laws or conic sections, and CAD/CNC users who need the exact perimeter to cut material efficiently all use this calculator, entering just the two semi-axis lengths.

Was this helpful?

You can pick up to 3 reactions.

Comments (1)

  • Editorial team

    To sanity-check a result, remember that the perimeter must fall between 2π times the semi-minor axis and 2π times the semi-major axis, because an ellipse sits between the circle that fits inside it and the one that encloses it. With 5 and 3 that window is 18.85 to 31.42, and 25.53 sits comfortably inside. If you are measuring edging for an oval bed or pond, add extra for overlaps and cuts. What shape are you outlining with it?

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators